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Astronomy, Astrophysics, and Cosmology Luis A. Anchordoqui Department of Physics and Astronomy Lehman College, City University of New York Lesson II February 16, 2016 arXiv:0706.1988 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 1 / 24 Table of Contents 1 Doppler Effect 2 Stellar Evolution L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 2 / 24 Doppler Effect Observational evidence that stars move at speeds ranging up to a few hundred kilometers per second E.g. + relatively fast moving Barnard’s star @ D ∼ 56 × 1012 km moves across line of sight @ v ∼ 89 km/s Consequence + proper motion shifts ∼ 0.0029◦ /yr HST has measured proper motions as low as about 1 marcs/yr In radio + VLBA relative motions can be measured to an accuracy of ∼ 0.2 marcs/yr Apparent position in sky of more distant stars changes so slowly that proper motion can’t be detected with even most patient observation However + rate of approach or recession of star in line of sight can be measured much more accurately than its ⊥ motion to line of sight Technique uses familiar property of any sort of wave motion Doppler effect L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 3 / 24 Doppler Effect Barnard’s star L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 4 / 24 Doppler Effect When we observe sound or light wave from source at rest time between arrival wave crests at our instruments is same as time between crests as they leave source However + if source is moving away from us time between arrivals of successive wave crests increases over time between departures from source because each crest has little farther to go on its journey to us than crest before Time between crests + wavelength divided by wave speed so a wave sent out by a source moving away from us will appear to have longer wavelength than source @ rest Likewise + if source is moving toward us time between arrivals of wave crests is decreased because each successive crest has shorter distance to go and waves appear to have shorter wavelength L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 5 / 24 Doppler Effect redshift and blueshift L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 6 / 24 Doppler Effect Weinberg’s analogy L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 7 / 24 Doppler Effect L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 8 / 24 Doppler Effect G Consider two inertial frames S and S0 moving with relative velocity v G Assume star @ rest in S0 emits light @ (ν0 , θ0 ) with respect to O0 G Momentum 4-vector for photon as seen in S hν hν hν µ p = , − cos θ, − sin θ, 0 (1) c c c G Momentum 4-vector for photon as seen in S0 hν0 hν0 hν0 µ p0 = ,− cos θ0 , − sin θ0 , 0 c c c L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology (2) 2-16-2016 9 / 24 Doppler Effect G Apply inverse LT to get 4-momentum relation from S0 → S hν0 hν0 hν = γ +β − cos θ0 c c c hν hν0 hν0 − cos θ = γ − cos θ0 + β c c c hν0 hν sin θ = sin θ0 c c (3) G First expression gives relativistic Doppler formula ν = ν0 γ(1 − β cos θ0 ) (4) G For observational astronomy (4) is not useful because (ν0 , θ0 ) refer to the star’s frame not that of observer G Apply instead direct LT S → S0 to photon energy ν0 = γν(1 + β cos θ ) (5) which gives ν0 in terms of quantities measured by observer L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 10 / 24 Doppler Effect Special cases θ0 = 0 + source moving away from the observer q ν = ν0 (1 − β)/(1 + β) (6) Non-relativistic limit + ν = ν0 (1 − β) θ0 = π + source moving towards observer q ν = ν0 (1 + β)/(1 − β) (7) θ0 = π/2 + transverse Doppler effect ν = ν0 γ (8) 2nd order relativistic effect arising from dilation of time in moving frame L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 11 / 24 Doppler Effect Search for exoplanets G As planet orbits star + star has its own orbit around CM system G Radial velocity method: Variations in star’s radial velocity detected from displacements in star’s spectral lines due to Doppler effect G Photometric method: If planet crosses in front of its parent star’s disk observed brightness drops by small amount L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 12 / 24 Stellar Evolution Stars appear unchanging Night after night heavens reveal no significant variations On human time scales + majority of stars change very little We cannot follow any but tiniest part of star life cycle L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 13 / 24 Stellar Evolution Star formation Stars are born when gaseous clouds (mostly hydrogen) contract due to pull of gravity Huge gas cloud fragments into numerous contracting masses Each mass is centered in area where density is only slightly greater than @ nearby points Once such “globules” formed gravity would cause each to contract in towards its CM As particles of such protostar accelerate inward their kinetic energy increases When kinetic energy is sufficiently high Coulomb repulsion + not strong enough to keep 1 H nuclei appart and nuclear fussion can take place In star like our Sun “burning” of 1 H occurs when 4p fuse to form 2 He nucleus with release of: γ, e+ , νe L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 14 / 24 Stellar Evolution M16 a.k.a. Eagle Nebula located ≈ 7, 000 ly away L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 15 / 24 Stellar Evolution HST’s pillars of creation L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 16 / 24 Stellar Evolution Sun’s energy output pp cycle due to following sequence of fusion reactions: 1 1 1H +1H →21H + 2 e+ + 2 νe (0.42 MeV) (9) 1 2 1H +1H →32He + γ (5.49 MeV) (10) (12.86 MeV) (11) 3 3 2 He +2He →42He +11H +11H Released energy ã mass difference between initial & final states â carried off by outgoing particles 4 11H →42He + 2e+ + 2νe + 2γ (12) Net effect Takes 2 of each of first 2 reactions to produce two 32 He Total energy released for net reaction + 24.7 MeV e+ quickly annihilates with e− to produce 2me c2 = 1.02 MeV so total energy released + 26.7 MeV Deuterium formation has very low probability infrequency of reaction limits rate at which Sun produces energy L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 17 / 24 Stellar Evolution In more massive stars... Energy output comes from the carbon (or CNO) cycle CNO cycle comprises following sequence of reactions: 12 1 6C +1H →137N + γ (13) →136C + e+ + ν (14) 13 1 6C +1H →147N + γ (15) 14 1 7N +1H →158O + γ (16) 13 7N 15 8O 15 1 7N +1H →157N + e+ +ν (17) →126C +42He (18) No carbon is consumed in this cycle (see first and last equations) Net effect is the same as the pp cycle Theory of the pp and CNO cyles first worked out by Bethe in 1939 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 18 / 24 Stellar Evolution Nuclear Burning L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 19 / 24 Stellar Evolution Fusion reactions take place in star core + T ∼ 107 K Surface temperature is much lower + O(few thousand K) Tremendous release of energy in these fusion reactions produces outward pressure to halt inward gravitational contraction protostar (now really a young star) stabilizes in main sequence Stellar structure on main sequence + described by spherically symmetric system in hydrostatic equilibrium This requires that rotation, convection, magnetic fields, and other effects that break rotational symmetry have only a minor influence on the star L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 20 / 24 Stellar Evolution G M(r ) + mass enclosed inside sphere with radius r and density ρ(r ) M (r ) = 4π Z r 0 2 dr 0 r 0 ρ(r 0 ) ⇒ dM (r ) = 4πr2 ρ(r ) dr (19) G Gravitational acceleration produced by M (r ) is g (r ) = − GM(r ) r2 (20) G If star is in equilibrium + acceleration balanced by pressure gradient from center of star to its surface G Since P = F/A + pressure change along distance dr yields dF = dAP − ( P + dP)dA (r )dAdr = − dAdP | {z } = − ρ | {z } force mass a (r ) |{z} (21) acceleration G For increasing r + gradient dP < 0 and resulting dF is positive and directed outward L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 21 / 24 Stellar Evolution G Hydrostatic equilibrium + g(r ) = − a(r ) GM(r ) ρ(r ) dP = ρ (r ) g (r ) = − (22) dr r2 G If the pressure gradient and gravity do not balance each other layer at position r is accelerated a (r ) = GM(r ) 1 dP + r2 ρ(r ) dr (23) G In general + equation of state P = P(ρ, T, Yi ) L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 22 / 24 Stellar Evolution G Estimate of central pressure + Pc = P(0) integrate (22) using (19) and obtain with P( R) ≈ 0 Pc = Z R dP 0 dr dr = G Z M dM 0 M , 4πr4 (24) G Pc lower limit + replace r by stellar radius R ≥ r Z M M 4 4πr 0 Z M M M2 > G dM = 4πR4 8πR4 0 Pc = G dM (25) G Inserting values for Sun M2 Pc > = 4 × 108 bar 8πR4 M M 2 R R 4 (26) G Integrating hydrostatic equation using “solar standard model” Pc = 2.48 × 1011 bar + factor 500 larger L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 23 / 24 Stellar Evolution to be continued . . . Literature: Longair: §8, 10, 12 & 13 HEA 09/ p L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 2-16-2016 24 / 24